Hurwitzquaternion

A Hurwitzquaternion (or Hurwitz integer) in mathematics is a quaternion, whose four coefficients either all ( rational ) integer or half-integer all ( halves of odd integers ) are - Mixtures of integers and half figures are therefore inadmissible. Is the set of all Hurwitzquaternionen

It forms in its quotient field, the division ring ( skew field ) of quaternions with rational coefficients

The ring of integral elements. is the smallest sub- body of the non- commutative Quaternionenschiefkörpers multiplication. On the other hand, its completion ( completion ) for the amount metric is straight again.

A Lipschitzquaternion (or Lipschitz integer ) is a quaternion, whose coefficients are all integers. The set of all Lipschitzquaternionen

Is a ( non- commutative ) subring of (but not ideal ). and have the same quotient field.

In contrast to is maximum as a whole ring and additionally a Euclidean ring, that is, knows a division with a small group and a Euclidean algorithm.

The article deals with the most important algebraic properties including symmetries of geometric and their effects. Further examples can be traced to what extent concepts, which are known from the commutative rings forth and are often defined only there, can be adapted for non-commutative setting.

The notes (note ) that are attached to a word or phrase, give explanations, details or examples.

Detached, comparisons with other dimensions, additional elaborations, etc. to the section in which they stand.

• 2.2.1 Lipschitz semigroup
• 2.2.2 Lipschitz units
• 2.2.3 Hurwitz semigroup
• 2.2.4 Hurwitz units

• 3.1 Regular 16- Zeller ( Hexadekachor )
• 3.2 Regular 8- Zeller ( tesseract )
• 3.3 Regular 24 - Zeller ( Ikositetrachor )

• 4.1 tiling with the 8- Zeller
• 4.2 tiling with the 24- Zeller
• 4.3 tiling with the 16- Zeller
• 4.4 spheres pack
• 4.5 stitches radius

Inheritances

The skew field "inherits" the, and all relevant rules of calculation of the quaternions with real coefficients. Regarding the definitions, please refer to the relevant article.

Is a 4 -dimensional vector space over its Skalarkörper as above. From vector space to win the addition and scalar multiplication, in which the scalar multiplying the quaternion component-wise. This multiplication is true in its domain of definition with the quaternion multiplication match because as is embedded in the quaternions, and it is commutative.

In this article, the ( full ) quaternion multiplication with the center point and the scalar multiplication is quoted by simple juxtaposition, and also the quaternions with Greek and the scalars are written with Latin letters.

Had to explain the impact of inheritances on the subject of the article and any quaternions ( with rational or real coefficients if necessary ).

• The scalar product defined by, is a positive definite symmetric bilinear form. We have the pictures and and images.
• The conjugation throws after.
• The norm is given by is ( the square of the amount), purely real, multiplicative and at a Hurwitzquaternion always a non-negative integer. According to the four- square theorem of Lagrange is needed more than 4 square numbers whose sum is for any non - negative integer. Thus, any non- negative integer is a standard Lipschitz (or Hurwitz ) quaternion.
• The positive definiteness of the scalar product means. It follows from the existence of the inverse   for, from the zero divisor of freedom.

Group Properties

The following notations are persevered in this article.

• The amount is additive and multiplicative closed because of the multiplicativity of the norm and subset of, as all have with an odd norm. Furthermore, for both and   as well.   is known as the lattice D4 [ref 1]. It is the straight " checksums " because even "checkerboard lattice ".
• Is a shorthand notation for the coset.
• The quaternion has the 6 -th power, and it is and.
• The amount is complete multiplicative.

Additivity

Lipschitz grid

The additive group is produced by, and forms a grating in the known as the grating I4 [ Ref 2].

Is a sublattice of index 2 of. This results in the partitions

Hurwitz lattice

As an additive group is free abelian with generators. also forms a lattice in, known as the F4 lattice [ Ref 3].

Is a sublattice of index 2 of and there are partitions

( see diagram below ). This is a complete set of representatives for.

The elements of the cosets have just the odd of " checksum ".

Multiplicativity

Lipschitz semigroup

It is clear that the product of two numbers Lipschitz with integer coefficients again has integral coefficients. Thus, the amount is a semigroup under the quaternion multiplication.

Lipschitz units

The unit group is the non- Abelian quaternion

Of order 8 with the center. Generating Q8 are, for example, and with the equations

Hurwitz semigroup

The proof of the multiplicative closedness of succeed without much calculation by assembling from the 4 cosets. [NB 1]

Conclusion: The quantities and are closed under addition and multiplication, such that they form ( non-commutative ) sub-rings in both their quotient field, and is an ideal in both rings (see also the section on ideals ).

Hurwitz units

Called the unit group, also the group Hurwitzeinheiten, is the non- Abelian group

Of order 24, which consists of the eight elements of the group Q8 and the 16 quaternions, where the signs are to be taken in any combination: the Hurwitzeinheiten in the strict sense. is isomorphic to the binary tetrahedral group 2T, a central group extension of the tetrahedral group T = A4 of order 12 with a cyclic group of order 2 is also your center and the factor group is isomorphic to A4.

Q8 is a normal subgroup of index 3 of, and is subset of with and; So [NB 2 ] is the semidirect product.

Are, for example be generated by

With the equations

Geometric properties

Regular 16- Zeller ( Hexadekachor )

The elements of the group Q8 all have the standard one and form the corners of the Kreuzpolytops the fourth dimension, the regular so-called 16- Zellers, also Hexadekachōr ( on) ( the, Eng. Hexadecachoron, from Greek ἑξαδεκάχωρον from hexa, six ' and deka, ten ' and Choros ' space' ) called. He is enrolled in the unit 3- sphere, which is a group yourself again, namely the Lie group SU (2). Its boundary consists of 16 tetrahedra with vertex sets, each of the 16 combinations of signs represents a tetrahedron. The centers of these tetrahedra are just the halves of the Hurwitzeinheiten in the strict sense.

The 16- Zeller is dual to the 8- Zeller, one of the six regular convex 4- polytopes ( Polychora in ), has Schläfli symbol { 3,3,4 } and is bounded by 16 ( regular ) tetrahedral cells, 32 ( regular ) triangular faces, 24 edges and 8 corners. His four - volume is at an edge length of and a beam radius of 1

Regular 8- Zeller ( tesseract )

The remaining 16 elements, d see the Hurwitzeinheiten in the narrow sense, also have the standard one and form the corners of the hypercube ( Maßpolytops ) of the fourth dimension, the so-called regular 8- Zeller, also known as the Tesseract. It is bounded by 8 cubes, one of which has, for example, the 8 corners and in the center. The midpoints of the cube are.

The 8- Zeller is dual to the 16- Zeller, one of the six regular convex 4- polytopes, has Schläfli symbol { 4.3.3 } and is bounded by 8 cells ( cubes ), 24 squares, 32 edges and 16 corners. His four - volume 1 is on an edge length and a radius of radius 1

Regular 24 - Zeller ( Ikositetrachor )

The elements of the group all have the standard one and form the corners of the so-called 24 - Zellers, also Ikositetrachōr ( on) ( the, Eng. Icositetrachoron, from Greek εἰκοσιτετράχωρον from eikosi, twenty ' and tetra, short form of τέτταρα, four ' Choros and ' space ' ), enrolled in the unit 3- sphere. The 6 quaternions mark the corners of a regular octahedron with the center on the edge of this 24 - Zellers, which merges with (left as right ) multiplication by an element in another octahedron ( on the edge ). Thus, the edge of the 24 Zellers of 24 ( regular ) is octahedral cells, which meet six at each corner and 3 on each edge. The 24- Zeller is one of the six regular convex 4- polytopes, has 24 cells ( the octahedron ), 96 triangular faces, 96 edges and 24 vertices. The 4- volume 2 is on an edge length and a radius of radius 1

The 24- Zeller has Schläfli symbol { 3,4,3 }, is the only self -dual regular Euclidean polytope that is not Simplex or polygon, and so far has no equivalent in other dimensions. [NB 3]

Tiling and packing spheres

Are available for each of the three above-mentioned regular 4- polytope is a regular and uninterrupted tiling - and these are the only [ Ref 4] - the 4-dimensional Euclidean space.

Tiling with the 8- Zeller

A tiling of the tesseract can be set up so that the centers of the tesseracts, the stitches fall exactly on the Lipschitzquaternionen. This is achieved with the above mentioned tesseract, more precisely, the 4-dimensional and quite open for the disjunction of the mesh interval than the basic stitch.

This tiling with the 8- Zeller is called the Lipschitz tiling. It has Schläfli symbol { 4,3,3,4 }, and is dual to itself, that is, the center points of a tiling are the vertices of the dual and vice versa. The 4- volume of the mesh is 1 when an edge length and a radius radius of 1 [NB 4]

Tiling with the 24- Zeller

A tiling of the leaves with the 24- Zeller be set up so that the centers of the 24 - Zeller fall exactly on the Hurwitzquaternionen. The basic stitch is the 24- Zeller with the center and the corners 24 of the species. [NB 5]

This tiling with the 24- Zeller is called the Hurwitz - tiling. Your Schläfli symbol is { 3,4,3,3 }. Is the 4- volume of the mesh with an edge length and a radius of radius. [NB 6]

Tiling with the 16- Zeller

There is a tiling with the 16- Zeller, which is the dual of the tiling with the 24- Zeller, - Schläfli symbol { 3,3,4,3 } So. The 4- volume of their mesh is at an edge length of 1 and a radius of radius. [NB 7]

Spheres pack

In connection with these latter two tilings is ( proved for lattice packings, but not for non -lattice packings [ Ref 5] ) packing density of 4- balls (3 - Spheres) on the Hurwitz lattice in F4 maximum. These spheres pack comes to a kiss number 24 ( the upper limit - even among non -lattice packings - proved [ Ref 6] ) [NB 8].

Mesh radius

For the division with remainder below we need the grid width of a grid and define it as the largest occurring distance

A point at a grid point which is closest to it, i.e.

The grid has the mesh radius. [NB 10]

Pseudo code for the approximation of a quaternion by a Lipschitz integer:

This is in mesh with the center, more precisely (right open 4 -dimensional interval) [NB 11].

The grid has the mesh radius. [NB 12]

Pseudo code for the approximation of a quaternion by a Hurwitz integer:

The standard deviation of the result. [NB 13]

[NB 14]

Euclidean

The following pseudo code is determined to be a left division with " small " radical the radical:

The name of the function is to suggest that the result of a left division stems and tentatively in a subsequenten multiplication as a left factor ( divisor ) occurs.

This division with remainder makes the ring of Hurwitzquaternionen to a right - Euclidean domain, that is, to two numbers and returns it with

As in commutative Euclidean rings is every ideal in a principal ideal - only must additionally laterality ( here first: right ) of the ideal are given [NB 16].

The following pseudo code shows an algorithm for finding an Euclidean left greatest common divisor ( gcd ) of two Hurwitzquaternionen in.

The result is a left divisor of and, that is, there is with and. And it also applies to the ( right side ) of Lemma Bézout, ie there is

The latter being obtained as a by-product of the Euclidean algorithm ( and can be led out of the function).

In the previous section you can see the two factors in each quaternion and everywhere at the terms "right" and "left" interchanged, and so it is also.

Thus, the ring is also left - Euclidean, that is, to two numbers and returns it with

And every left ideal in a left principal ideal.

Conclusion: is two-sided Euclidean - or Euclidean par excellence.

Automorphisms

As an automorphism of an algebraic structure applies a bijective mapping, in which all algebraic links are treated homomorphic, that is, for example,

The prime field of skew field must remain fixed. In contrast, the three imaginary units ( the quaternion group Q8 produce ) can be converted into a each other. The automorphisms of Q8 can all continue to automorphisms of ( unique). The sub-groups and inherit these automorphisms by restriction. Thus, the automorphism, and isomorphic to the rotation group and the octahedron, which in turn is isomorphic to the symmetric group S4.

The automorphisms can be realized by (used in) "inner" automorphisms:

The factor group has 24 elements and is thus isomorphic to the automorphism groups discussed here ( and the symmetric group S4).

The conjugation as a reflection in the real axis is involutive, and ( as with Q8) antihomomorph [ Ref 7] in the multiplication, ie

And is therefore called involutional Antiautomorphismus.

Associated elements

The concept of mutually associated elements can be used for non-commutative rings somewhat broader: 2 elements and associated extended to each other when there are two units. There is to a maximum of Hurwitzquaternion 242/2 = 288 expands Associated, as the whole group must be run on the other only the factor group modulo the center on either side. The Associated awareness is an equivalence relation as in the commutative case.

If, then either or (see Hurwitz lattice ), that is, at any Hurwitzquaternion there is left ( just right) associated Lipschitzquaternionen.

The conjugate is not normally associated.

Ideal

The Hurwitzquaternionen make an order (in the sense of ring theory ) in its quotient field, the division ring ( skew field ) of quaternions with rational coefficients. They are there even a maximal order or wholeness ring. The Lipschitzquaternionen - as at first glance, closer candidate for the concept of whole quaternions - also represent a fine, but are not maximal and have no division with a small rest is why they are less suitable for the development of a theory of ideals, algebraic with the number theory would be comparable. Adolf Hurwitz has recognized this - a big step in the theory of maximal orders. Another was the finding that - in a non - commutative ring like - not unique are (all purely imaginary Einheitsquaternionen have squared), so that one to a set must be provided when the concept of algebraic integer to skew field wants to transmit.

For with, therefore, is the automorphism of a (outer ) automorphism of. The left ideal is equal

Therefore right ideal, ie two-sided and the same for all these 24 generators. Furthermore, it is a maximal ideal with factor ring isomorphic to the finite field of characteristic 2 whose multiplicative group is isomorphic to and containing the 3 -th primitive roots of unity (see addition and multiplication table ). is as a maximum in a factor ring. [NB 17]

Prime elements factorization

A Hurwitzquaternion is prime in if and only if its norm is in prim.

The following peculiarities of natürlichzahligen ( purely real ) Hurwitzquaternionen are in the context of Primelementzerlegung of concern:

Each Hurwitzquaternion can be decomposed into prime factors, the order of the prime factors can be specified in the following sense: Be a Hurwitzquaternion and

A decomposition into prime factors of their standard. Then for any sequence of these prime factors of a decomposition of

In primes in with

For a given prime number sequence, the factorization is up to units between the prime elements, or left and right of it and the many possibilities of splitting natural divider (this is a prime number must be in the prime number sequence is at least 2 times occur ) clearly. For the factorization in several algorithms are available. One of the prime number in corresponding prime element in one can, for example, find those with the function described above and then split off just left. If the result of, then the prime occurs in the prime number sequence at least two times before, and you can select any prime element among their many Jacobi splittings. [NB 19]

Metric completion and power series expansion

Archimedean valuation and metrics

The "natural" Evaluation of the skew field is the sum Rating

Since each variable can be overtaken in magnitude by multiplying a unit size, this review [NB 20] is called Archimedean. , This amount induces the metric

Exactly the Euclidean distance in the corresponding. It is well known, satisfies the axioms for metrics:

The completion of the metric leads to the quaternions with real coefficients. The completion of the metric leads to nothing new, since a discrete subset of being.

For every Hurwitzquaternion a unique representation by each of the two finite series

With the base [ Ref 8], digits, base powers the right or left of it and one with.

This place value system, which can be extended to the whole, has the following properties:

• It comes without a " sign " from.
• The presentation is almost everywhere clearly reversible. [NB 21]
• The Hurwitzquaternionen exactly correspond to the illustrations without a decimal.
• The elements, and only these rational elements have periodic representations. [NB 22]

Nichtarchimedische evaluation and metrics

For a fixed prime number is for each

The exponent of the standard. This ( exponent ) Rating met:

Note that the exponent of the standard is not met in a prime condition ( C). [NB 23] That at work, is due to the two-sidedness of the ideal.

It is a group of " units "

Assign to which there is a valuation ring. [NB 24]

The valuation ring is

A local ring with the valuation ideal ( maximal ideal )

Where the ( scalar ) standards, the denominator of contributing. The connection to the 2 - adic rational integers creates for the equation

The distance function defined by

Also satisfies the axioms for metrics. Add to that the

( 4) aggravated triangle inequality,

Which makes it an ultrametric. The completion of this metric leads to

Quaternions with 2- adic coefficients. The completed valuation ring is

Coincides with the completion of the ring of Hurwitzquaternionen, because sealing is [NB 26]. This is the unique continuation of on.

The completed evaluation is ideal

Where with, and the residue field is isomorphic to the mentioned in Section ideals.

If we take as a complement operator, we obtain the diagram on the left for the completions of the cosets of which, in contrast to the above, however, are no more grid.

As with the p- adic numbers we have, at a fixed prime element, a unique -adic representability of an element through each of the two convergent series

With (see above system of representatives ) and powers of the base left or right. The elements, and only these rational elements have periodic representations. [NB 27]

[NB 28]

Comments ( Note)

• (3, 2, 0, 0 ) · ( 3, -2, 0, 0) = (2, 2, 2, 1 ) · ( 2, -2, -2, -1 )

• (1, 1, 1, 0 ) * ( 3, -2, 0, 0 ) · ( 3, 2, 0, 0)
• (1, 1, 1, 0 ) · ( 2, -2, -2, -1 ) · ( 2, 2, 2, 1)

• (3, 2, 0, 0 ) · ( 0, 1, 1, -1) · (2, -2, -2, 1)
• (2, -2, -2, -1 ) · ( 0, 1, 1, 1 ) * ( 3, 2, 0, 0)
• (2, 2, 2, 1 ) · ( 1, -1, -1, 0 ) · ( 2, 2, 2, -1)

• (3, 2, 0, 0 ) · ( 3, -2, 0, 0 ) · ( 1, 1, 1, 0)
• (2, 2, 2, 1 ) · ( 2, -2, -2, -1 ) * ( 1, 1, 1, 0)

The selection criterion for the numbers is one of the divisibility - ie a non- archimedean. Therefore, the numbers are in ascending order delivered from the low to the high powers of. The potencies are right here - hence the function name - factors of digits. To fit the left-to- right descending Horner scheme, e.g.. Usually, in Archimedean value systems, the terms with exponents ≥ 0 are left; Decimal places close to the right of optional value - separator "" to indicating in its asymmetry, the direction in which the exponent can be infinite, thus marks the difference between and Archimedean and non- Archimedean representation. ( For finite representations that direction does not matter. ) The attached subscript expresses that the base is, and that their powers to the right of the coefficients (numbers) are located. Thus we have, for example.

A genuinely archimedean selection criterion for the digits, which supplies the descending digits of the high potencies and is suitable for all, is much more complicated to formulate, as you can tell by the shape of the adjacent kite-shaped area. It shows the uniquely achievable by a power in the Gaussian plane through sums of magnitude lower area in the same shades of gray - at each power step, an exact duplicate of the Association of has gone before is added.

The additive system of generators has the resp. Encodings. ( The codes are of the type that are powers of the base are so right. The delimiter is omitted in whole numbers. ) Hereby and by means of the addition table shown can encode any Hurwitzquaternion. ( However, addition tables are skillful, containing for each of the 9 digits sum of two summands and each of the 225 possible transfers the new transfers for the construction of an arithmetic for Hurwitzquaternionen. )

Due to need of the multiplication table, only the squares

A mention.

With the commutation digits of the right to be placed on the left side of. Since and are in the center, we have and, so that the table can be limited to 3 entries.

Additional coding examples:

The following function can encode the non- archimedean context of all quaternions with arbitrary precision, even those who have no finite codes (hence the addition of the maximum number of digits against the Archimedean function).

Suitable coding systems ( and) ( your results differ, not only in the number sequence, but possibly also in the finite. ) With prime element in and of order 3 or 6 Are for a nichtarchimedische and Archimedean encoding (with the same system ) are both finite, then match the codes.

• For all odd numbers, so the sum of 1-4 odd squares. Thus, there is no four squares in which non- trivial sum, so contains no zero divisors - as it should be.
• Even at 5 summands it looks quite different: All require at least four summands, so that their squares form the sum. On the other hand, is a quadratic residue, so there is a with. For now was such that, even. Then, that is, there is a - convergent sequence whose square is. With the result that the sum of the squares 5 disappears. All or need at least three summands, so that their squares form the sum. On the other hand, are neither quadratic residues, so it needs at least 2 additional numbers and. Can be chosen as described above so as to disappear.
• From each of these results follows further that it (the list is complete) in the quaternion algebras with coefficients from the above quadratic fields, and zero divisors are.
• Zero divisor is also among the quaternions with coefficients from the over square body because of the sum of squares.

References (Ref)